This invention relates to a single sideband transmitter apparatus for translating input signals to other frequency bands.
As is well known, there are substantial difficulties in implementing single sideband transmitters. This is because the common conceptual versions of the single sideband transmitter either require that a relative 90.degree. phase shift be introduced across the entire baseband signal or that there be an ideal rectangular bandpass filter characteristic provided on one side or the other of the carrier frequency to filter off the unwanted sideband. This filter can be less than ideal if the baseband signal does not have zero and low frequency components contained therein such as may be found with voice information signals or with data channels using certain partial response techniques. Nevertheless, there are often difficult requirements to be met and the filters in these circumstances must still have excellent characteristics for operation in a high frequency band for filtering off the unwanted sideband of a translated baseband signal.
FIG. 1 shows a prior art system using a bandpass filter to eliminate the unwanted sideband. A signal e.sub.s is used to modulate a carrier in a product device, a balanced modulator, and the resulting translated signal having two sidebands, the intermediate signal i.sub.2 (t), is supplied to a bandpass filter to eliminate one of the sidebands. FIG. 2 shows the sequence of steps along the Fourier frequency axis in a series of magnitUde versus frequency diagrams.
Frequency axis A in FIG. 2 shows the arbitrarily assumed frequency content of the signal e.sub.s. Skipping frequency axis B for the moment, frequency axis C shows the representation of the carrier wave e.sub.c (t), a Dirac delta function or frequency impulse in the frequency domain. Since e.sub.s (t) is multiplied with the carrier e.sub.c (t) in the time domain as indicated in FIG. 1 to provide i.sub.2 (t), the representations on frequency axes A and C are to be convolved with one another in the frequency domain to produce the results shown on frequency axis D. The intermediate signal represented along frequency axis D is applied to the idealized rectangular filter (represented by the dotted lines along frequency axis D) and the result is shown on frequency axis E where the lower sideband has been arbitrarily retained.
FIG. 3 shows the well known 90.degree. phase shift method of generating a single sideband output signal. Again, the magnitude versus frequency plots of FIG. 2 can be used to show the frequency axis effects of the operations shown in FIG. 3. The frequency content of the input signal e.sub.s (t) is again that shown on frequency axis A and the 90.degree. phase shifted input signal, i.sub.1 (t), is shown on frequency axis B. Both carrier waves e.sub.c (t) and e'.sub.c (t) can be represented by the frequency impulses on Dirac delta functions shown on frequency axis C. The two carrier waves represented on frequency axis C are convolved with the signals shown on frequency axes A and B, respectively, with the results of each convolution shown on frequency axis D corresponding to intermediate signals i.sub.2 (t) and i.sub.3 (t), where the idealized filter characteristic is now to be ignored. Both signals can be so represented since no phase information is included in these magnitude versus frequency plots.
By combining the two convolution results shown on frequency axis D in FIG. 2 by the summer shown in FIG. 3, the output signal results are obtained, e.sub.o (t) and the output signal frequency content is shown on axis E, again the lower sideband being arbitrarily retained. The position of the carrier wave frequency if present, would be that shown by the dotted-in impulses along frequency axis E. They are not present given the transmitter shown but are usually inserted at the transmitter for demodulation purposes at the receiver.
The input signal in both of the foregoing cases has been shown with no zero frequency component and the sidebands in the baseband signals could be pulled back some from zero frequency to ease the filtering required in the filtered version, a limitation which may be satisfactory in a voice channel where voice frequency components near zero are negligible. Also, the frequency axis of the input signal, axis A, has been labelled with some arbitrary frequencies for purposes of comparison with later Figures and these frequencies are intended to apply to the frequency axes set out below frequency axis A.
FIG. 4 represents an extended version of FIG. 3 in that a sampled data source is assumed rather than an analog source which led to the diagrams of FIG. 2. The sample data source provides samples at the rate of 1/T samples per second to form e.sub.s (kT). Assuming that an ideal sampling situation exists, an ideal lowpass filter having a rectangular characteristic with a cutoff of 1/(2T) hertz can be used to recover the analog signal which was sampled to provide the signal e.sub.s (kT). The signal at the output of the ideal lowpass filter will then be e.sub.s (t). The remainder of FIG. 4 is then that of FIG. 3. An analysis of the system of FIG. 4 in the time domain is of interest for later comparison and is set out as follows to give the derivation of the output signal.
By use of the Dirac delta function .delta.(t-t.sub.o), the sequence of input data samples e.sub.s (kT) can be written in terms of a continuous time parameter as follows: ##EQU1## This signal is applied to the ideal low pass filter having a rectangular characteristic along the Fourier frequency axis with a cutoff frequency of 1/(2T) hertz. Such a filter has an impulse response as follows: ##EQU2##
The result at the output of the filter is e.sub.s (t): ##EQU3##
This result is multiplied by the first carrier wave in the multiplier provided in the upper leg of the second section of the transmitter in FIG. 4 to provide intermediate signal i.sub.2 (t): ##EQU4## where the last equation is obtained by using the following trigonometric identity: EQU 2cos x sin y - sin (x+y) - sin(x-y)
As is well known, phase shifting a signal by 90.degree. can be accomplished by taking the Hilbert transform of the signal. The pertinent Hilbert transform in this situation is the following: ##EQU5##
Thus, taking the Hilbert transform of e.sub.s (t) provides the second intermediate signal to be applied to the second multiplier, i.sub.2 (t): ##EQU6##
The second multiplier in the lower leg of the remainder of the transmitter multiplies the quadrature carrier wave, e'.sub.c (t), with i.sub.1 (t) to provide the third intermediate signal, i.sub.3 (t) giving the following result: ##EQU7##
The two intermediate signals, i.sub.2 (t) and i.sub.3 (t) are then combined in the summer shown in FIG. 4 to provide the output signal e.sub.o (t). The results is shown below: ##EQU8##
This can be rewritten as follows: ##EQU9##
This analysis shows a form that can be taken by a single sideband transmitter in providing an analog output signal with a sample data input signal.
The earlier discussions indicated that often an ideal 90.degree. relative phase shift apparatus is required to provide an acceptable single sideband transmitter or, alternatively, that excellent bandpass filters passing frequencies near the carrier wave frequency are required. The difficulty of implementing these kinds of devices makes use of a single sideband transmitter difficult and a single sideband transmitter avoiding such devices insofar as possible is therefore quite desirable.